The Text

1845 Constitutional Amendment:
No judicial officer shall be removed by the joint resolution of the two houses of the Legislature, or by the Senate on the recommendation of the Governor, unless the cause of such removal shall be entered on the journal of both houses, or of the Senate, as the case may be ; and such officer against whom the Legislature or the Senate may be about to proceed, shall be served with notice thereof, accompanied with a copy of the causes alleged for his removal, at least twenty days before the day on which either house shall act thereupon, and shall have an opportunity to be heard in his defence before any question shall be taken upon such removal; and the yeas and nays shall be entered upon the journals of the Senate, or House, as the case may be.

The full provision includes both the amendment and the text it amends – NY Constitution 1822, Article I, Section 13:
All officers holding their offices during good behavior may be removed by joint resolution of the two houses of the legislature, if two thirds of all the members elected to the assembly, and a majority of all the members elected to the senate, concur therein.